The Chain Polynomial Of Two Variables

Let　　L be a link, the sign P_L ( l ,m ) denotes the polynomial with two invariants corresponding to it. In this article, Firstly we remember the properties of the P_L ( l ,m ) and prove these properties simply, in the same time, we discuss P_L ( l ,m ) with two new splitting patterns; Secondly using these splitting patterns and the computation of P_L ( l ,m ), we advance a new link type S -link, then discuss its properties, finally we give the estimation of breadth of the P_L ( l ,m ) which corresponding to a S-link: assume that L is a S -link. Then span(l) ≤c^* (L) + |S_C1^*| + |S_C2^*|2, span(m)*(L)+c(L)-|S_C^*|. where S_C1^*, S_C2^* are obtained by smoothing exactly C₁^* positive crossings and C₂^* negative crossings ; Thirdly we give some examples of S-link, showing the existence of S -link; At last, we give a distinguishing methods: for a link L , if there are states {S_C^* }:1. compute sum from S P. 2. let l = it^-1, m = i (t^1/2-t^-1/2), finding ν_L^'(t) corresponding to sum form S P. 3. compare ν_L^'(t) with ν_L(t) in the table , then draw a conclusion .